Four tyres and a spare
A car has 4 tyres in use and 1 spare, all identical. Each tyre wears out after exactly 20,000 km of use. You may swap tyres as often as you like.
What is the greatest distance, in km, the car can travel?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Count the total tyre life available, and how much the car uses per kilometre.
Can the rotation be arranged so that all five tyres wear out at the same time?
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Idea. Count in tyre-kilometres: the five tyres together offer a fixed budget, and the car spends 4 per kilometre. Rotating so that every tyre is used equally spends the whole budget.
Step 1: the bound. The five tyres provide tyre-km. Each kilometre uses 4 tyre-km. So the car can travel at most km.
Step 2: reaching it. Split the trip into 5 legs of 5,000 km, and rest a different tyre on each leg. Each tyre runs on 4 legs, km, and all five wear out together at 25,000 km.
So the answer is 25,000 km.